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[Sorry if the level here is wrong, I asked this on math.SE, but even with a bounty, it got no attention.]

I am currently reading Hatcher's 3-Manifolds notes, the part proving Alexander's theorem, which is a specific case of the generalized Schoenflies theorem:

Every smoothly embedded $S^2\subset \mathbb{R}^3$ bounds a smooth 3-ball.

The proof seems to rely on intuition for these low-dimensional arguments, which I find disconcerting, because I have not yet developed that intuition, so I am trying to give actual formal proofs for the statements in Hatcher's proof.

The proof begins with a generic smoothly embedded closed surface $S\subset\mathbb{R}^3$. I have been able to prove that I can isotope $S$ so that projection on the last coordinate $\pi:\mathbb{R}^3\rightarrow\mathbb{R}$ is a Morse function on $S$. Hatcher then argues that if $t$ is a regular value for $\pi$, then $\pi^{-1}(t)\cap S$ is a finite collection of circles.

The proof continues by taking an innermost circle $C\subset \pi^{-1}(t)\cap S$, which by 2-dimensional Schoenflies bounds a disk $D$, and $D\cap S=\partial D=C$. Hatcher then uses surgery to cut away a neighborhood of $C$ in $S$, and cap the cuts with two disks.

This last part is what I want to formalize. It seems we are finding a small-enough tubular neighborhood $C\times(-\epsilon,\epsilon)\subset S$, and then removing that, leaving $S_-=C\times\{-\epsilon\}$ and $S_+=C\times\{\epsilon\}$. Again by 2-dimensional Schoenflies, these bound disks $D_-$ and $D_+$. What I don't get is: why are $S_-$ and $S_+$ still innermost? Or, put differently, why is $D_-\cap S=S-$ (and similarly for $S_+$)?

Intuitively, this seems obvious, and it seems like some sort of "continuity" argument would work, but I cannot figure out how to make this formal. I tried proving that in fact all the disks, "stacked" together for the tubular neighborhood, gave a smooth $D\times [-\epsilon,\epsilon]$, but again I find it hard to make topological arguments when one step of the construction is "apply Schoenflies to get a disk". In particular, I can't prove the projection of this "solid neighborhood" to $D$ is continuous.

Does anyone know how to formalize this? Or, even better, a reference where this type of surgery is discussed? I checked a few places, but only found surgery on a single manifold, not the type discussed here, where we're surgering an embedded submanifold in some ambient manifold.

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    $\begingroup$ Hatcher's notes assumes (roughly) that the reader is familiar with the proof of the h-cobordism theorem and the main content of Milnor's book on Morse Theory book. The style and conventions in the types of arguments he's using are built-up when studying those topics. $\endgroup$ – Ryan Budney Oct 20 '20 at 0:54
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    $\begingroup$ @RyanBudney: do you think reading through Milnor's books would give me the background to formalize this argument? Is there some key step or convention that I'm missing ? $\endgroup$ – Hempelicious Oct 20 '20 at 14:42
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    $\begingroup$ Maybe Guillemin and Pollack's "Differential Topology" would be the more appropriate place to start. The transversality (stability) arguments there would help you with this particular argument. $\endgroup$ – Ryan Budney Oct 20 '20 at 18:40
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For an interval $[a,b]\subset{\mathbb R}$ in which the height function $f:S\to {\mathbb R}$ has no critical values one obtains a product structure on $f^{-1}([a,b])$ by following flow lines of the gradient vector field of $f$. This vector field on $f^{-1}([a,b])$ can be extended to a vector field on ${\mathbb R}^2\times [a,b]$ with positive $z$-coordinate everywhere in ${\mathbb R}^2\times [a,b]$. To do this one can first extend the gradient vector field on the surface to a tubular neighborhood of the surface via a projection of this neighborhood onto the surface, then use a smooth partition of unity subordinate to the cover of ${\mathbb R}^2\times [a,b]$ by the tubular neighborhood and the complement of the surface to combine the vector field on the neighborhood with the vertical vector field $(0,0,1)$ on the complement of the surface. In formulas the combined vector field would have the form $v=\phi_1 v_1+\phi_2 v_2$ where $v_1$ is the vector field on the neighborhood and $v_2$ is the vector field on the complement of the surface, with the partition of unity functions $\phi_1$ supported in the neighborhood and $\phi_2$ supported in the complement of the surface. The flow lines of this extended vector field $v$ then give a new product structure on ${\mathbb R}^2\times [a,b]$ extending the product structure on $f^{-1}([a,b])$. In other words one has a level-preserving diffeomorphism of pairs $({\mathbb R}^2\times [a,b],f^{-1}([a,b]))\approx ({\mathbb R}^2\times [a,b], f^{-1}(a)\times [a,b])$.

This is a special case of the isotopy extension theorem which says that an isotopy of a submanifold can always be extended to an ambient isotopy of the whole manifold. The proof is essentially the same.

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    $\begingroup$ I love that on this site you can ask for help with a textbook’s exposition, and get an answer directly from the textbook’s author. If this answer doesn’t merit the bounty, I can’t imagine what would! $\endgroup$ – Rivers McForge Oct 24 '20 at 2:13
  • $\begingroup$ Thank you! Extending things to the ambient slice is the piece I was missing. $\endgroup$ – Hempelicious Oct 24 '20 at 14:59
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If $t$ is a regular value, then it is a property of Morse functions that there is some small open neighborhood $U$ of $t$ in $\mathbb{R}$ such that $u$ is also a regular value for all $u\in U$. In particular, we can take $U=(t-\delta,t+\delta)$ for some $\delta>0$. But then $\pi^{-1}(U)\cap S\cong(\pi^{-1}(t)\cap S)\times U$, ie. the surface is a product between any two successive critical levels. Now simply choose $\epsilon<\delta$. Then $S_-=(C\times U)\cap\pi^{-1}(t-\epsilon)$ and $S_+=(C\times U)\cap\pi^{-1}(t+\epsilon)$ will be innermost since $C$ is innermost.

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  • $\begingroup$ Sorry, I don't follow the final sentence at all. Why would there be a critical value? $\endgroup$ – Hempelicious Feb 3 '20 at 3:54
  • $\begingroup$ I deleted that sentence. Is it clear now? $\endgroup$ – Josh Howie Feb 3 '20 at 4:55
  • $\begingroup$ I think it's intuitively clear, and does help me understand things better, but I'm trying to avoid intuition. In particular, your diffeomorphism with the product does not take into account the ambient slices, so I don't see why we can say anything about a circle being innermost or not. Here's an example of what I'm worried about: if all we know at each slice is there are two circles, there's nothing a priori preventing one moving "inside" another. Intuitively, that can't happen because the enclosed disks would change how they intersect. It's this last part I can't formalize. $\endgroup$ – Hempelicious Feb 3 '20 at 19:35
  • $\begingroup$ That cannot happen because there is also a product $D\times U$. $\endgroup$ – Josh Howie Feb 3 '20 at 22:21
  • $\begingroup$ I think that's the Crux of the issue: how do we know that? How do we know the disks we get from Jordan curve theorem can be aligned as a product via the map you've provided? $\endgroup$ – Hempelicious Feb 3 '20 at 23:52
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You have a collection of circles which move smoothly (hence homotopically, see below) with the height (due to regularity and implicit function theorem). Let us define what means "innermost": It means that if you take a point $x$ from the inner circle then every outer circle will have winding number 1 or -1 around $x$. Or if you take a point $x$ from an outer circle then the inner circle will have winding number 0 around $x$. Now simply use the fact that the winding number is invariant under homotopies, that is, if $H\colon[a,b]\times[0,1]\to\mathbb R^2$ and $x\colon[0,1]\to\mathbb R^2$ are continuous with $H(t,0)=H(t,1)$ and such that $x(t)\notin H(\{t\}\times[0,1])$ for all $t\in[a,b]$ then the winding number $n(H(t,\cdot),x(t))$ is independent of $t\in[a,b]$.

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